Journal of Geophysical Research: Atmospheres

Condensation of water vapor: Experimental determination of mass and thermal accommodation coefficients

Authors


Abstract

[1] Experimental determinations of mass and thermal accommodation coefficients αm and αt for condensation of water vapor in air have been conducted covering a temperature range from about 250 to 290 K. For the first time, both coefficients have been determined directly and simultaneously. To this end, growth of water droplets in air has been observed at different total gas pressures ranging from about 1000 down to 100 hPa. Monodispersed seed particles have been used as condensation nuclei. After addition of water vapor with well-defined partial vapor pressure, supersaturation was achieved by adiabatic expansion in an expansion chamber. Most experiments reported in the present paper were performed at vapor saturation ratios ranging from 1.30 to 1.50. Monodispersed Ag particles with a diameter of 9 nm have been used as condensation nuclei, and for humidification a diffusion humidifier was applied. One experiment was performed at the saturation ratio of 1.02, which resembles conditions observed in the Earth's lower atmosphere. In this experiment, monodispersed DEHS particles with a diameter of 80 nm were used as condensation nuclei, and water vapor was generated by quantitative evaporation of a liquid jet. Droplet growth was monitored using the CAMS method. For determination of the accommodation coefficients, experimental droplet growth curves were compared to corresponding theoretical curves. Quantitative comparison was performed by varying the respective accommodation coefficient and the starting time of droplet growth in a two-parameter best fit procedure. Considering the uncertainty with respect to the starting time of droplet growth and the uncertainties of the experimental water vapor supersaturation, corresponding maximum errors have been determined. From the results obtained it can be stated that αt is larger than 0.85 over the whole considered temperature range. For 250–270 K, values of αm below 0.8 are excluded, and for higher temperatures up to 290 K we can exclude values of αm below 0.4. Both coefficients are likely to be unity for all studied conditions. The results of this study enable accurate predictions of the formation and growth of cloud droplets required to parameterize cloud light scattering/absorption and precipitation properties in climate models.

1. Introduction

[2] Atmospheric aerosol particles affect the life on Earth in various ways. In polluted urban environments aerosol particles degrade visibility and, if inhaled, influence human health [Peters and Wichmann, 2001; Stieb et al., 2002; Cabada et al., 2004]. Aerosol particles also affect the Earth's radiative forcing and consequently the global climate both directly and indirectly. The direct effect on the radiative balance is mainly due to absorption and scattering by the aerosol particles. On the other hand, the indirect effect is connected to changes in cloud formation caused by aerosol particles acting as cloud condensation nuclei, thereby leading to changes in planetary albedo and hydrological cycle [Charlson et al., 1992; Ramanathan et al., 2001; Lohmann and Feichter, 2005]. For this reason knowledge on the sources of atmospheric aerosol particles, particularly on their formation and growth [e.g., Kulmala et al., 2004], is essential. Also investigations of the physical processes relevant for the formation and growth of cloud droplets are required to determine e.g. cloud droplet concentrations [Kulmala et al., 1996], and are therefore needed for accurate climate system modeling.

[3] The stability of clouds and their ability to produce precipitation are strongly influenced by the number density and size distribution of their constituent water droplets. The activation of aerosol particles (cloud condensation nuclei, CCN) and their subsequent growth into cloud droplets are clearly related to the transfer of water vapor molecules into liquid droplets. The growth of cloud droplets and aqueous atmospheric aerosol particles is controlled by both heat and mass transfer. Water vapor fluxes and the resulting heat transfer are particularly important. In the continuum regime gas phase diffusion and thermal conductivity are the controlling phenomena. In the kinetic regime mass accommodation coefficient αm and thermal accommodation coefficient αt are the two parameters that fundamentally influence the interaction of water vapor or any other gaseous molecule with a liquid surface [Winkler et al., 2004]. The mass accommodation coefficient is the probability for a vapor molecule striking the liquid-gas interface to stay within the liquid, whereas the thermal accommodation coefficient is the probability for a gas molecule striking the liquid-gas interface to come into thermal equilibrium with the liquid before being diffusively reflected. A proper description of formation and growth of cloud droplets requires detailed knowledge of vapor, particularly water vapor transport as well as heat transport in the vicinity of atmospheric aerosol particles and cloud droplets. Because of its importance, the mass accommodation coefficient of water vapor has been the subject of at least 40 published experimental studies over the past 75 years, while only few experimental studies exist on the thermal accommodation coefficient. The results of these studies [Cammenga, 1982; Li et al., 2001; Marek and Straub, 2001; Mozurkewich, 1986; Shaw and Lamb, 1999] range over three orders of magnitude for the mass accommodation coefficient and approximately one order of magnitude for the thermal accommodation coefficient showing the lack in reliable data.

[4] Although the most important application of thermal as well as mass accommodation coefficients is related to growth or evaporation of water droplets in the atmosphere, these coefficients have so far not been determined simultaneously over a sufficient temperature range. In the present paper we report direct determinations of mass accommodation coefficient of water vapor on water and thermal accommodation coefficient of mainly air on water at different temperatures from quantitative comparison of experimental and theoretical droplet growth curves. As described below, measurements at different ambient total gas pressures allowed simultaneous determinations of both mass and thermal accommodation coefficients. Error analysis of all relevant experimental parameters is of particular importance. We expand on the study previously reported by Winkler et al. [2004; see also Davidovits et al., 2004] by presenting all the experimental data and a detailed error analysis on the experimentally determined accommodation coefficients. We also give a thorough description of the experimental and theoretical backgrounds as well as the fitting procedure used for simultaneous determination of the accommodation coefficients.

2. Experimental Approach

[5] In the present study, both thermal and mass accommodation coefficients for water vapor condensation in air have been determined simultaneously from quantitative comparison of experimental and theoretical droplet growth curves. Measurements were performed at temperatures ranging from about 250 to 290 K and at different total gas pressures from about 1000 down to 100 hPa. The investigated water saturation ratios ranged from 1.02 to 1.50. In most experiments liquid droplets nucleated on nearly monodispersed Ag seed particles and growing due to condensation of supersaturated water vapor have been observed using the experimental system shown schematically in Figure 1.

Figure 1.

Schematic diagram of the experimental system for determination of accommodation coefficients.

[6] The apparatus includes a source of electrostatically classified, monodisperse particles and a vapor saturation unit. Vapor supersaturation was achieved by adiabatic expansion in the computer controlled thermostated expansion chamber of the Size-Analyzing Nuclei Counter (SANC), resulting in well-defined thermodynamical conditions in the measuring volume. Droplet growth was observed using the Constant-Angle Mie Scattering (CAMS) detection method [Wagner, 1985].

[7] Particles were generated by evaporation and subsequent condensation of Ag. Carefully dried and filtered air was passed through a tube furnace with a flow rate of 1.8 l/min. Evaporation of Ag in the furnace at 1333 K and subsequent homogeneous nucleation leads to the formation of polydispersed Ag particles. At the exit of the tube furnace dried and filtered air was added at a flow rate of 1.7 l/min in order to minimize coagulation of the newly formed Ag particles. Subsequently, the Ag particles were bipolarly charged in an Am241 charger. Using an electrostatic classifier a nearly monodispersed and unipolarly charged particle fraction was extracted. This fraction was brought into charge equilibrium in an Am241 neutralizer in order to avoid the possible effect of surface charges on the outcome of the condensation experiments. The size distribution of the Ag particles obtained was monitored by an electrical mobility spectrometer. Nearly lognormal size distributions were found with a mean particle diameter of 9.0 nm and a geometric standard deviation of 1.06. Particle number densities between about 1 and 1.5 × 104cm−3 were obtained.

[8] Water vapor was subsequently added to the system by evaporation from a thermostated cylindrical liquid film inside the vapor saturation unit. A sufficiently long residence time in the saturator was allowed in order to establish a water vapor partial pressure very close to equilibrium at the saturator temperature. During a computer controlled measurement cycle the saturated binary vapor-air mixture together with the Ag particles was passed into an expansion chamber. Sufficiently long flushing time periods were chosen in order to provide equilibrium conditions at the walls of the expansion chamber. The expansion chamber was thermostated typically 1 to 5 K above the saturator temperature. Accordingly, well defined initial conditions with partial vapor pressures slightly below saturation were obtained.

[9] In order to achieve the desired vapor supersaturations, adiabatic expansion was initiated by opening a valve connecting the chamber to a low-pressure buffer tank. The total gas pressure before expansion has been measured by means of a pressure gauge PG, whereas the pressure drop inside the expansion chamber was monitored by a fast precision pressure transducer. Temperature and vapor saturation ratio after expansion were determined using Poisson's law. Vapor saturation ratios ranging from 1.30 to 1.50 were obtained. It is notable that the volume of the buffer tank is by a factor of about 25 larger than the volume of the expansion chamber and therefore the total gas pressure in the chamber remains practically unchanged after expansion has occurred.

[10] Particle generator, classifier and spectrometer have continuously been operated at atmospheric pressure, however, appropriate setting of a precision metering valve at the inlet of the vapor saturation unit allowed condensation experiments to be performed over a range of initial total gas pressures from about 1000 hPa down to 100 hPa.

[11] Ag particles causing heterogeneous nucleation at the vapor supersaturations considered will lead to formation and growth of liquid droplets. Growth rates and number densities of the condensing droplets were measured using the CAMS detection method [Wagner, 1985]. To this end the droplets growing in the cylindrical expansion chamber were illuminated by a laser beam. The light flux transmitted through the expansion chamber as well as the light flux scattered at a selectable, fixed scattering angle were monitored during droplet growth. The light scattering arrangement with lens-pinhole detection is shown schematically in Figure 2.

Figure 2.

Schematic diagram of the light scattering arrangement for CAMS measurements.

[12] In order to avoid a possible influence of light extinction on the scattered light flux, particularly at elevated droplet number densities, the scattered light flux has been normalized with respect to the light flux transmitted through the expansion chamber. A typical experimental normalized scattered light flux versus time curve is shown together with the corresponding theoretical normalized scattered light flux versus droplet radius curve according to Mie theory in Figure 3.

Figure 3.

Typical normalized scattered light flux measured as a function of time (top curve) and normalized scattered light flux calculated as a function of size according to Mie theory (bottom curve). Scattering angle is Θ = 15°, and refractive index is n = 1.333. τ indicates the time interval during which the adiabatic expansion occurs in the expansion chamber.

[13] Satisfactory agreement of the rich morphology of experimental and theoretical light scattering curves is observed allowing to establish a unique correspondence of experimental and theoretical light scattering extrema. Quantitative comparison of the two curves then enables independent determinations of droplet number densities and droplet radii during droplet growth. Comparison of the height of corresponding experimental and theoretical light scattering maxima yields the number density of the growing droplets. Independently, from the position of the experimental light scattering extrema on the time axis, droplet radii are obtained at different times during the growth process. Furthermore, detailed analysis of the first light scattering maximum allows to extend the size measurement range toward smaller droplet radii. It is notable that the CAMS method provides absolute, time-resolved and noninvasive simultaneous determinations of droplet diameter and number density. No influence of the laser beam on droplet condensation has been found. The time interval τ shown in Figure 3 indicates the period of time, during which the adiabatic expansion occurs in the expansion chamber.

[14] It is difficult to determine the actual starting time for droplet growth, however, we have experimentally limited the starting time to a time interval Δt, within which the observed droplet growth is expected to start. The end of this starting time interval Δt can be obtained from detailed analysis of the pressure versus time curve shown in Figure 4 as the position where the pressure reaches the absolute minimum corresponding to a vapor saturation ratio Shigh. Beyond this minimum no further nucleation will occur.

Figure 4.

Total gas pressure in the expansion chamber measured as a function of time. The definition of the experimental starting time interval Δt for droplet growth is illustrated (see text for additional details).

[15] In order to obtain the beginning of the time interval Δt the minimum vapor saturation ratio Slow needs to be determined (see Figure 4), below which no significant heterogeneous nucleation by the Ag particles considered is observed. In the present experiments a relatively low vapor saturation ratio S0 was typically chosen so that droplets with only comparatively small number density N0 were nucleated just sufficient to allow CAMS measurements. Accordingly, below a saturation ratio Slow resulting in a droplet number density of N0/2, the respective scattered light fluxes are already too small to allow proper evaluation and thus no significant nucleation is observed. For determination of Slow the relation between number density of nucleated droplets and corresponding saturation ratio is required. This relation has been established in separate experiments providing the heterogeneous nucleation probability for the Ag particles as a function of the water vapor saturation ratio as shown in Figure 5. A detailed study on heterogeneous nucleation probabilities has been presented recently [Wagner et al., 2003]. As can be seen from Figure 5, the experimental saturation ratio S0, which corresponds to the droplet number density N0, is typically below the onset saturation ratio Sonset, where 50% of all Ag particles are nucleated.

Figure 5.

Experimental heterogeneous nucleation probability for nucleation of water vapor on 9 nm Ag particles. The definitions of Sonset and Slow are illustrated (see text for additional details).

[16] Slow can now be obtained as the saturation ratio corresponding to the droplet number density N0/2 (see Figure 5). Finally, from the pressure versus time curve during adiabatic expansion, as shown in Figure 4, the beginning of the time interval Δt is determined as the time corresponding to the vapor saturation ratio Slow.

[17] In addition to the above mentioned experiments we report an additional droplet growth experiment at the comparatively small saturation ratio 1.02 [Rudolf, 1994], which is resembling conditions observed in the Earth's lower atmosphere. In this experiment di-ethyl-hexyl-sebacate (DEHS) particles with a diameter of 80 nm obtained from a collision atomizer were used as condensation nuclei. Water vapor generation was performed by quantitative evaporation of a liquid jet. An expansion chamber with inner diameter 286 mm and volume 35 l provides a sufficiently long sensitive time. A detailed description of this version of the Size-Analyzing Nuclei Counter (SANC II) has been presented elsewhere [Rudolf, 1994; Rudolf et al., 2001].

3. Theoretical Approach

[18] In this study the theory of droplet growth has extensively been used for determination of the accommodation coefficients. Both the mass and thermal accommodation coefficients play the role of adjustable parameters in droplet growth theory. In the following the basic features of droplet growth theory will be described. The theory of droplet growth in unary vapor systems has been investigated by several authors [Fuchs, 1959; Sedunov, 1974; Wagner, 1982]. Droplet growth is controlled by both mass and heat transfer, which are in fact coupled. Condensing vapor molecules produce latent heat, which has to be transported away from the droplets. Under the experimental conditions considered it can be assumed that mass and heat fluxes are quasistationary. Consequently, vapor concentration and temperature profiles always correspond to steady state solutions and changes during the growth process are determined only by the changing boundary conditions. The processes of mass and heat transfer are different in the kinetic, transition and continuum regimes. These regimes are usually characterized by the Knudsen number Kn = λ/a, where λ is the mean free path in the surrounding gas and a is the droplet radius. For large Knudsen numbers (λa) droplet growth takes place in the kinetic (or free molecule) regime, whereas for small Knudsen numbers (λa) growth is in the continuum regime. For particle sizes in the range of the mean free path (Kn ≃ 1), growth is in the transition regime.

[19] Solution of the diffusion equations for vapor concentration and temperature yields the corresponding profiles, which satisfy the boundary conditions at the droplet surface and at infinity. Subsequently, the continuum mass flux Ic directed away from a single droplet can be obtained in the form [Kulmala and Vesala, 1991; Vesala et al., 1997]

equation image

where a is the droplet radius, Mv is the molecular weight of the vapor, p is the total pressure, X is the vapor mole fraction far from the droplet and Xa is the mole fraction just above the liquid surface. T is the temperature far away from the droplet. D designates the binary diffusion coefficient of the mixture of vapor and inert gas at temperature T. C takes into account the temperature dependence of D more rigorously than the commonly used geometric mean. Because of the temperature dependence of equilibrium vapor pressure and diffusion coefficient Ic is not only a function of the droplet radius a but also of the temperature Ta at the droplet surface.

[20] The continuum heat flux Qc directed away from a single droplet can be expressed as [Vesala et al., 1997]

equation image

where Ka and K correspond to thermal conductivities of the binary mixture of inert gas and vapor at the droplet temperature (Ta) and the gas temperature far from the droplet (T), respectively. Hv is the specific enthalpy of the vapor. The heat exchange by radiation is ignored, which is a good assumption for the experimental conditions considered. Note that Ic and its temperature dependence now couples equations (1) and (2).

[21] Generally, the transport of mass and energy is partly under kinetic control. The combined effects of kinetic and continuum transport can be accounted for by applying appropriate corrections to the continuum fluxes. The mass and heat flux in the transition regime can be expressed as

equation image

and

equation image

where Ic is the continuum mass flux. βM is the transition regime correction for mass transfer given by [Fuchs and Sutugin, 1970]

equation image

and βT is the transitional correction factor for heat transfer given by

equation image

[22] It can be seen that βM and βT are dependent on the accommodation coefficients αm and αt and on the Knudsen numbers KnM = λv/a and KnT = λg/a. λv is the mean free path of the vapor molecules calculated from the vapor diffusivity D, λg is the mean free path of the gas molecules obtained from the thermal conductivity K of the inert gas [Wagner, 1982]. Clearly in this study αm refers to water vapor molecules, while αt refers (mainly) to inert air molecules. For very small Knudsen numbers the transitional correction factors βM and βT approach unity and accordingly the fluxes become independent of the accommodation coefficients. This is consistent with the fact that in the continuum regime the droplet growth process is under diffusion control and surface kinetic effects have no significant influence. Accordingly, measurements of the accommodation coefficients cannot be performed in the continuum regime.

[23] As shown by Qu and Davis [2001], the transition regime corrections used in the present study in conjunction with the proper definitions of the Knudsen numbers [Fuchs and Sutugin, 1970] provide good approximations for molecular mass ratios ranging from values ≪ 1 (light vapors) up to values as much as 10 and even higher.

[24] The growth rate of a droplet can be determined from the total transitional mass flux to the droplet which can be calculated from equations (1) and (3). In order to obtain the total transitional mass flux as a function only of a, Ta has to be accounted for. This temperature can be obtained from the heat balance equation [Vesala et al., 1997]

equation image

at the droplet surface which relates mass and heat flux. Hl(Ta) is the specific enthalpy of the liquid at the droplet temperature Ta. After inserting QT and IT according to equations (1), (3), (2) and (4)Ta can then be obtained as a function of droplet radius by numerical solution of equation (7). Inserting Ta into (1), (3), the mass flux IT(a) can be calculated for one single droplet.

[25] Finally, an expression describing the evolution in time of the droplet radius is required. Calculation of the quasi steady state growth rate of the droplets can be performed according to the equation

equation image

where mD = equation imageπ a3ρL is the mass of a spherical droplet, ρL being the liquid density at temperature Ta. Integration of equation (8) yields the growth time t as a function of the droplet radius a:

equation image

where a0 is the initial droplet radius.

[26] In the present study the simultaneous growth in a (nearly monodispersed) population of droplets was observed. Accordingly, changes of partial vapor pressure and ambient temperature caused by depletion of vapor and production of latent heat during droplet growth had to be quantitatively accounted for at the integration in equation (9). Numerical inversion of the function t(a) finally yields the droplet radius a(t) as a function of time and the droplet growth calculation is completed. The physico-chemical parameters used in the present calculations are listed in Table 1.

Table 1. Physico-Chemical Properties of Water and Aira
Physico-Chemical PropertyValue
  • a

    R, general gas constant; Mg, molecular mass of air; Mv, molecular mass of water; κg, adiabatic index of air; κv, adiabatic index of water; cp,g, specific heat capacity of air at constant pressure; cp,v, specific heat capacity of water at constant pressure; Cvg, molar heat capacity of air at constant volume; ρ, density of water; σ, surface tension of water; L, specific latent heat of vaporization; D, binary diffusion coefficient; Kg, thermal conductivity of air; Kv, thermal conductivity of water vapor; K, thermal conductivity of water vapor air mixture; ps(T), equilibrium vapor pressure of water; T, temperature [K]; p, total gas pressure [Pa]; pv, partial pressure of water vapor; pg, partial pressure of air.

  • b

    Lindsay and Bromley [1950].

  • c

    Wukalowitsch [1958].

R8314.7, equation image
Mg28.96456, equation image
Mv18.016, equation image
equation imageg1.402
equation imagev1.343
cp,g1005, equation image
cp,v1860, equation image
Cvg(T)20465.3+1.3017T, equation image
ρ(T)1049.572-0.1763T, equation image
σ(T)0.117296-0.152362 · 10−3T, equation image
L(T)3.14566·106-2361.64T, equation image
D(T, p)1.9545·10−4T1.6658p−1, equation image
Kg(T)3.4405·10−3+7.5177 · 10−5T, equation image
Kv(T)−6.7194·10−3+7.4857 · 10−5T, equation image
K(equation image, T)equation image + equation image,bequation image
lnps(T)77.34491296-7235.424651T−1 −8.2ln(T) + 0.0057113T, ps in Pac

[27] From the time dependence of the droplet radius the ambient parameters as well as the droplet temperature can be calculated as functions of time. Figures 6 and 7show the numerically obtained time dependence of droplet temperature, ambient (gas) temperature and ambient saturation ratio calculated for different droplet concentrations. While temperatures and saturation ratio stay practically constant for the case of one single droplet, considerable changes are observed at higher concentrations. This shows clearly the influence of droplet concentration on temperature and saturation ratio.

Figure 6.

Droplet temperature and gas temperature during condensational growth in droplet populations with number concentrations 1 and 4381 cm−3 calculated as functions of time.

Figure 7.

Saturation ratio during condensational growth in droplet populations with number concentrations 1 and 4381 cm−3 calculated as a function of time.

[28] The ambient gas temperature T is found to increase toward a limiting value. Similarly, the vapor saturation ratio S decreases toward the value 1. It is notable, on the other hand, that the droplet temperature Ta remains approximately constant during the entire growth process even at higher droplet concentrations. Accordingly, in the present experiments droplet growth was observed at well-defined and approximately constant droplet temperatures. The influence of droplet concentration on droplet growth rates is shown in Figure 8. For higher concentrations droplet growth rates are found to be somewhat reduced at the later stages of droplet growth, which is due to vapor depletion and production of latent heat. As can be seen from Figure 8, the corresponding experimental data are in good agreement with the theoretically calculated growth curve indicating that vapor depletion and production of latent heat are properly accounted for in the present numerical calculations.

Figure 8.

Droplet growth curves calculated for condensational growth in droplet populations with number concentrations 1 and 4381 cm−3. Corresponding experimental data are shown for comparison.

4. Simultaneous Determination of Mass and Thermal Accommodation Coefficients

[29] Growth rates and number densities of condensing water droplets are measured using the CAMS-method as described earlier. Comparison of experimental and theoretical droplet growth curves yields quantitative information on mass accommodation coefficients αm and thermal accommodation coefficients αt.

[30] As described in the previous section, droplet growth is determined by the fluxes of mass and heat in the vicinity of the droplets. These fluxes are depending on the binary diffusivity D of vapor in air and the thermal conductivity K of the mixture of air and vapor, respectively. With decreasing total gas pressure p the diffusivity D is increasing while the thermal conductivity K remains practically constant. Accordingly, it can be expected that at sufficiently reduced total gas pressures the heat flux is the rate controlling process for droplet growth, while droplet growth will not be significantly limited by the mass flux. Therefore in this case the droplet growth rate will be increasingly sensitive with respect to the thermal accommodation coefficient αt and practically independent of αm. On the other hand, however, with increasing pressure p the diffusivity D is decreasing. Accordingly, both heat and mass fluxes will be simultaneously controlling the droplet growth process. Therefore at these conditions it can be expected that the droplet growth rates will depend on both the thermal accommodation coefficient αt and the mass accommodation coefficient αm.

[31] The above described behavior can be further explained on the basis of droplet growth theory. By elimination of the droplet temperature Ta using the heat balance equation (7) an approximate analytical expression for the transitional mass flux IT directed away from a single droplet can be obtained [Wagner, 1982; Kulmala et al., 1989; Fladerer et al., 2002]. From this analytical expression for IT the relation

equation image

can be derived considering constant system temperature. This relation provides information on the sensitivity of mass flux and thus of droplet growth rate with respect to changes of the accommodation coefficients. To a good approximation K/D is proportional to the total gas pressure p. Accordingly, it can be concluded from the relation (10) that for decreasing total gas pressure p droplet growth is becoming increasingly sensitive with respect to changes of αt as compared to the sensitivity of droplet growth on αm.

[32] On the basis of the full numerical solution for the mass flux IT a numerical sensitivity analysis was performed. We have actually found that at a total gas pressure around 900 hPa the theoretical droplet growth curves are about equally sensitive with respect to changes of αm and αt. This is illustrated in Figure 9 showing theoretical growth curves at a total gas pressure of about 900 hPa for different values of mass and thermal accommodation coefficients αm and αt. However, as seen from Figure 10, at a reduced total pressure of about 180 hPa the growth curves are far more sensitive with respect to αt as compared to αm. Accordingly, at the reduced total pressure the thermal accommodation coefficient αt can be determined practically independent of αm. Subsequently, using the just obtained value of αt, comparison of experimental and theoretical droplet growth curves at a total gas pressure around 900 hPa allows a unique determination of the mass accommodation coefficient αm.

Figure 9.

Theoretical droplet growth curves at a total gas pressure of 903.2 hPa calculated for different values of mass and thermal accommodation coefficients. Corresponding experimental data are shown for comparison.

Figure 10.

Theoretical droplet growth curves at a total gas pressure of 182.20 hPa calculated for different values of mass and thermal accommodation coefficients. Corresponding experimental data are shown for comparison.

5. Data Evaluation and Error Analysis

5.1. Quantitative Determination of Accommodation Coefficients

[33] In order to obtain values for the accommodation coefficients, experimental droplet growth data obtained by the CAMS method were compared to theoretical droplet growth curves. We obtained droplet growth data from series of subsequent growth measurements at unchanged conditions. Identification of extrema of the averaged experimental light scattering curve by comparison to the corresponding theoretical Mie scattering curve yielded the experimental droplet growth times at respective droplet radii. On the other hand, from analysis of each individual experimental light scattering curve the standard deviations σ of the experimental droplet growth times were determined. To data points where statistical evaluation was not possible, a value of σ = 1 ms was attributed. Uncertainties of the droplet radii were not considered since inaccuracies of the droplet radii according to Mie theory are negligible.

[34] For quantitative evaluation of the accommodation coefficient α (αt or αm) we performed a comparison of experimental and theoretical droplet growth data, where the starting time of droplet growth as well as α were varied in a two-parameter fit procedure accounting for uncertainties of the experimental data [Vrtala, 2002]. Applying a χ2-test, the absolute minimum of Σequation image was searched, where x-xt is the difference between experimental and theoretical droplet growth time and σ denotes the standard deviation of the experimental droplet growth time x. The search for the minimum has been performed by splitting Σequation image into Σ+equation image and Σequation image, where Σ+ and Σ account for the positive (x-xt ≥ 0) and negative (x-xt < 0) deviations in time, respectively. This allows to obtain a direction for searching the minimum. Finding the position where Σ+ and Σ are equal and absolutely minimal yields α as the best value. The two-parameter fit procedure is carried out using a bisection method for determination of α after an initial screening process ruling out local minima. For each value of α the time interval Δt, as determined from analysis of the pressure drop curve, is scanned in sufficiently small steps in order to find the best position of the theoretical drop growth curve with respect to the time axis. Being more efficient in computing time than sophisticated multidimensional minimization algorithms this direct search method has been found to be appropriate.

[35] In order to determine the maximum error of α for a fixed value of the saturation ratio S, one-parameter fits were performed, where the starting time of droplet growth is fixed to the left or the right end of time interval Δt and only α is varied. Theoretical growth curves, which are assumed to pass through the left or right limit of the experimental time interval Δt, yield lower or upper limits of the accommodation coefficient α, respectively.

[36] Figures 1113 show typical experimental droplet growth data including the experimental time interval Δt. Furthermore, theoretical growth curves corresponding to two-parameter and one-parameter fits are indicated referring to thermal and to mass accommodation coefficient. The two-parameter fit indicates a start of the growth process within the experimental time interval Δt. This has generally been found for all growth data obtained in this study. Accordingly, the position of the experimental time interval Δt is found to be consistent with the fit to the droplet growth data and does therefore not influence the fit value of α. On the other hand, the length of Δt allows the determination of the maximum error range of α corresponding to one fixed value of the vapor saturation ratio S.

Figure 11.

Droplet growth curves corresponding to different fit values of αt at a total pressure of 179.8 hPa. Droplet temperature is 278.5 K, and initial saturation ratio is 1.596. Solid line indicates two-parameter fit, and dashed lines indicate one-parameter fits. Corresponding experimental data are shown for comparison.

Figure 12.

Droplet growth curves corresponding to different fit values of αm at a total pressure of 903.2 hPa. Droplet temperature is 279.1 K, and initial saturation ratio is 1.378. Solid line indicates two-parameter fit, and dashed lines indicate one-parameter fits. Corresponding experimental data are shown for comparison.

Figure 13.

Droplet growth curves corresponding to different fit values of αm at a total pressure of 894 hPa. Droplet temperature is 278.3 K, and initial saturation ratio is 1.020. Solid line indicates two-parameter fit, and dashed lines indicate one-parameter fits. Corresponding experimental data are shown for comparison. Note the comparatively slow droplet growth process.

5.2. Quantification of Errors in the Determination of Experimental Vapor Supersaturation

[37] Droplet growth curves are generally quite sensitive with respect to the vapor saturation ratio. Accordingly, accurate determination of experimental uncertainties of the vapor saturation ratio is important for evaluation of uncertainties of accommodation coefficients. The vapor saturation ratio S in the expansion chamber after adiabatic expansion is determined by temperatures of humidifier THum and expansion chamber TCha, initial total gas pressure before expansion P and total pressure drop during expansion dP. Determination of the uncertainty of vapor saturation ratio was performed applying the Gaussian law of error propagation

equation image

where ΔS1 to ΔS4 denote the differences of vapor saturation ratio at corresponding variation of chamber temperature ΔTCha, humidifier temperature ΔTHum, total pressure ΔP and pressure drop ΔdP, respectively. For the uncertainties of THum, TCha, P and dP we used the values σTHum = ±0.05 K, σTCha = ±0.05 K, σP = ±2 hPa, and σdP = ±0.5 hPa, respectively. Performing calculations at total gas pressures of 1000 hPa and 200 hPa, the above mentioned uncertainties result in an uncertainty of S of ±0.01 and ±0.03, respectively.

5.3. Sensitivity of Theoretical Droplet Growth Curves With Respect to Thermodynamical Data

[38] As can be seen from equations (1) and (2) theoretical droplet growth is mainly depending on diffusion and thermal conductivity. Accordingly, appropriate choice of the corresponding transport coefficients from literature is essential for accurate calculations. We have used the binary diffusion coefficient D listed in Table 1. The influence of D on theoretical droplet growth curves has been tested by varying this value within ±2% which spans all data found in literature. In Figure 14 theoretical growth curves accounting for the 2% deviations of D are illustrated together with experimental data. As can be seen, the observed changes are negligible.

Figure 14.

Theoretical droplet growth curves calculated for different values of the binary diffusion coefficient. Corresponding experimental data are shown for comparison.

[39] On the other hand, the thermal conductivities Kg and Kv of inert gas and vapor, as listed in Table 1, were compared to literature data. Variations of Kg by ±1% and Kv by ±5% were found to be appropriate and corresponding theoretical growth curves are illustrated in Figure 15. Again, no significant changes are observed. Consequently, uncertainties of the transport coefficients used in the present study will not be considered in the determination of errors of the accommodation coefficients.

Figure 15.

Theoretical droplet growth curves calculated for different values of the thermal conductivities Kg and Kv. Corresponding experimental data are shown for comparison.

6. Results and Discussion

[40] Accommodation coefficients α (αt or αm) were obtained by comparison of the experimental droplet growth data to corresponding theoretical droplet growth curves, where the starting time of droplet growth as well as α were varied in a two-parameter best fit procedure. We have performed a set of experiments covering a range of droplet temperatures from about 252 to 290 K. Average water vapor phase activities (saturation ratios) S0 were chosen ranging from 1.30 to 1.50 just appropriate to generate droplets with number concentrations between about 3000 and 4000 cm−3 as required for quantitative light scattering evaluations. During measurements at humidifier temperatures below 273 K we were accounting for the fact that the liquid film inside the humidifier was frozen. The observed light scattering curves clearly indicate, however, that the growing water droplets stay liquid even at the lowest temperatures considered. The saturation vapor pressure over ice was obtained according to Flatau et al. [1992] and Rasmussen [1978].

[41] In order to check a possible influence of the vapor supersaturation on the mass accommodation coefficient one experiment was performed at an initial vapor saturation ratio of 1.02, which is resembling conditions observed in the Earth's lower atmosphere.

[42] Combination of the maximum error corresponding to uncertainties in the starting time for drop growth at fixed saturation ratio with the error caused by the uncertainty of the saturation ratio results in the maximum total error of the accommodation coefficient α. In Tables 2 and 3 the corresponding values of αm and αt are listed.

Table 2. αm at Corresponding Temperaturesa
T, KαmRange
  • a

    For the measurements, the total gas pressure was ranging from 841 to 926 hPa.

252.11.120.9–1.31
258.11.20.86–1.78
259.30.970.83–1.33
265.81.180.84–2.05
270.21.090.79–1.73
271.60.910.6–1.37
278.30.960.42–4.26
279.11.030.56–1.85
289.50.830.41–7.37
289.90.860.44–16.26
Table 3. αt at Corresponding Temperaturesa
T, KαtRange
  • a

    For the measurements, the total gas pressure was ranging from 180 to 182 hPa.

258.71.030.88–1.32
278.50.960.88–1.13
287.60.980.86–1.14

[43] The independently obtained experimental accommodation probabilities αt (mainly for air) and αm (for water) are shown as functions of the droplet temperature in Figures 16 and 17. The maximum total errors are indicated. The error ranges for αm show a tendency to increase with increasing droplet temperature. This is related to a decrease of the sensitivity of droplet growth curves with respect to αm with increasing droplet temperature. Particularly the large error ranges at about 290 K are notable. The experiment at 278.3 K corresponds to the initial vapor saturation ratio of 1.02. The somewhat increased range of maximum total errors for this experiment is related to the increased sensitivity of the growth rates with respect to uncertainties of the vapor saturation ratio in this range of comparatively small supersaturations. Of course, the actual values of accommodation probabilities cannot exceed unity.

Figure 16.

Thermal accommodation coefficient αt shown as a function of the droplet surface temperature. Maximum total errors are indicated.

Figure 17.

Mass accommodation coefficient αm shown as a function of the droplet surface temperature. Maximum total errors are indicated. The value at temperature 278.3 K was obtained from experiment at initial saturation ratio 1.020, and all other values correspond to initial saturation ratios between 1.30 and 1.50.

7. Conclusions

[44] Accommodation coefficients α (αt or αm) were determined by comparison of experimental droplet growth data to theoretical droplet growth curves, where starting time of droplet growth as well as α were varied in a two parameter fit procedure. Liquid droplets nucleated on insoluble Ag particles, 9 nm diameter, or on insoluble DEHS particles, 80 nm diameter, and growing due to condensation of supersaturated water vapor have been studied for a temperature range from about 250 to 290 K and total gas pressures of about 1000 and 200 hPa. Most experiments were performed at initial vapor saturation ratios between 1.30 and 1.50, one experiment corresponds to an initial saturation ratio of 1.02 resembling conditions observed in the Earth's lower atmosphere. It was found that the theoretical growth curves are about equally sensitive with respect to the mass accommodation coefficient αm at both total pressures, while the growth curves are considerably more sensitive with respect to the thermal accommodation coefficient αt at the reduced pressure of about 200 hPa. This allowed independent determinations of thermal and mass accommodation coefficients. Accounting for the uncertainty in the starting time of experimental droplet growth and the uncertainties of the experimental supersaturation, maximum total errors for αt and αm have been determined.

[45] The experimental data on thermal and mass accommodation coefficients presented in this paper [see also Winkler et al., 2004] result in definitive lower limits of both thermal and mass accommodation coefficients for condensation of water vapor in air over a considerable temperature range. It can be stated that values of αt<0.85 are excluded over the whole temperature range considered. For temperatures from 250 to 270 K values of αm < 0.8 are excluded, for higher temperatures up to 290 K we can exclude values of αm < 0.4. It is notable that all data are consistent with αm and αt equal to 1. This result resolves long-standing open questions on these values. The physical importance of our results is clear: No empirical factors are required anymore to describe heat and mass fluxes to or from a water droplet at well defined conditions. Also the atmospheric relevance is evident: Simulations of cloud droplet formation and growth using the model described by Kulmala et al. [1996] and assuming the mass accommodation coefficient of water vapor to be below 0.1 show that for most atmospheric conditions relevant to cloud formation water vapor saturation in nascent clouds would increase so much that a much larger fraction of aerosol particles would activate as cloud droplets [Rudolf et al., 2001]. This would lead to more stable clouds with a higher number density of smaller droplets, that is, clouds that are less likely to form precipitation. On the other hand, the above mentioned cloud activation simulations show that cloud droplet nucleation and growth rates are relatively insensitive to variations in αm above 0.5. The values of the accommodation coefficients presented in the present paper allow accurate predictions of the formation and growth of cloud droplets required in climate and meteorological models in order to correctly parameterize cloud light scattering/absorption and precipitation properties.

[46] Besides water vapor, also other vapors will participate the formation and growth of atmospheric aerosols and cloud droplets. Although the information on the accommodation coefficients for these vapors is limited, more and more information is being obtained: for example for sulphuric acid [see, e.g., Jefferson et al., 1997; Pöschl et al., 1998; Hanson, 2005] and nitric acid [Rudolf et al., 2001] vapors in water mixture the mass accommodation coefficients seem to be near unity. However, in the future more well-defined laboratory experiments are needed to figure out accommodation and uptake coefficients for atmospherically [see, e.g., Kulmala and Wagner, 2001] relevant compounds.

Acknowledgments

[47] This work was supported by the Austrian Science Foundation (project P16958-N02) and by the Hochschuljubiläumsstiftung of the City of Vienna. The Academy of Finland is acknowledged for its support.

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